Description of laminar-turbulent transition of an airfoil boundary layer measured by differential image thermography using directed percolation theory

Transition from laminar to turbulent flow is still a challenging problem. Recent studies indicate a good agreement when describing this phase transition with the directed percolation theory. This study presents a new experimental approach by means of differential image thermography (DIT) enabling to investigate this transition on the suction side of a heated airfoil. The results extend the applicability of the directed percolation theory to describe the transition on curves surfaces. The experimental effort allows for the first time an agreement between all three universal exponents of the (1+1)D directed percolation for such airfoil application. Furthermore, this study proves that the theory holds for a wide range of flows, as shown by the various conditions tested. Such a large parameter space was not covered in any examination so far. The findings underline the significance of percolation models in fluid mechanics and show that this theory can be used as a high precision tool for the problem of transition to turbulence.

Sharpness of the Phase Transition for the Orthant Model

The orthant model is a directed percolation model on $\mathbb {Z}^{d}$ ℤ d , in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.

Critical behavior of density-driven and shear-driven reversible–irreversible transitions in cyclically sheared vortices

Random assemblies of particles subjected to cyclic shear undergo a reversible–irreversible transition (RIT) with increasing a shear amplitude d or particle density n, while the latter type of RIT has not been verified experimentally. Here, we measure the time-dependent velocity of cyclically sheared vortices and observe the critical behavior of RIT driven by vortex density B as well as d. At the critical point of each RIT, $$B_{\mathrm {c}}$$ B c and $$d_{\mathrm {c}}$$ d c , the relaxation time $$\tau $$ τ to reach the steady state shows a power-law divergence. The critical exponent for B-driven RIT is in agreement with that for d-driven RIT and both types of RIT fall into the same universality class as the absorbing transition in the two-dimensional directed-percolation universality class. As d is decreased to the average intervortex spacing in the reversible regime, $$\tau (d)$$ τ ( d ) shows a significant drop, indicating a transition or crossover from a loop-reversible state with vortex-vortex collisions to a collisionless point-reversible state. In either regime, $$\tau (d)$$ τ ( d ) exhibits a power-law divergence at the same $$d_{\mathrm {c}}$$ d c with nearly the same exponent.

A directed percolation model for the permeability of young sea ice

<p>The permeability of sea ice is an important property with regard to the role of sea ice in the earth system. It controls fluid flow within sea ice, and thus affects processes like desalination and melt pond drainage. It also impacts the role of sea ice in hosting sea ice algae and organisms, and the uptake and release of nutrients and pollutants from Arctic surface waters. However, as sea ice permeability is difficult to measure in the field, observations are sparse and vary, even for similar porosity, over orders of magnitude. This range is related to the evolution of the sea ice pore space during aging from young ice to thick first year ice. In young ice, the pore network is dominated by primary pores constrained by brine layers and the near-interface microstructure. In older sea ice, the ongoing desalination and thermal fluctuations have created wider secondary brine channels, implying a several orders of magnitude higher permeability. It is a challenge to understand and model these changes in pore space and permeability. Here a directed percolation model for the permeability of young sea ice is proposed. The model describes the dependence of sea ice permeability and electrical conductivity on brine porosity, and its critical behaviour and percolation transition due to necking of pores, and related disconnection of pore networks. Its parameters are based on 3D X-ray micro-tomographic imaging of young sea ice and direct numerical simulation of its transport properties, that strongly support the application of directed percolation theory to young sea ice, with a threshold porosity (impermeable ice) of 2 to 3 percent. Combined to an approach to predict the crystal structure at the ice-ocean interface, the model also the growth-velocity dependence and evolution of permeability near the ice-ocean interface. As the model is strictly valid for growing and cooling sea ice, without present of wider secondary brine channels, it is mostly relevant for sea ice desalination processes during winter. Modelling permeability of older and summer ice (and melt pond drainage) will require more observations of the pore space evolution in warming sea ice, for which the present results can be considered as a starting point.</p>